Journal of the
Korean Mathematical Society JKMS

We study the set of critical exponents of discrete groups acting on regular trees. We prove that for every real number $\delta$ between $0$ and $\frac{1}{2}\log q$, there is a discrete subgroup $\Gamma$ acting without inversion on a $(q+1)$-regular tree whose critical exponent is equal to $\delta$. Explicit construction of edge-indexed graphs corresponding to a quotient graph of groups are given.

Keywords: groups acting on trees, critical exponents, Ihara zeta function

MSC numbers: Primary 20E08; Secondary 05E18, 57M60